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Molecular Modelling Lecture Notes
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Module-3
Ab Initio Molecular Dynamics
March 18
Molecular Dynamics (MD) Simulation
Ab initio MD
PLAN
R. Car and M. Parrinello Phys. Rev. Lett. 55, 2471, 1985
Molecular Dynamics
We like to move....atoms!How can I do it?
The Story of Newtons Apple...Equations of motion:
Solving (integrating) D.E.
F = md2x
dt2
We may use numerical or analytical integration
trusting truly on the
classical equations of
motion
F=ma
Mangalyaan Mission
also used to create your favorite characters!
gradient
how to get forces?
Fi = @U(x1, x2, , xn)@xi
Born-Oppenheimer Molecular Dynamics
U(RN ) = minD |Hel|
E+ ENN
{i}
Ab Initio Molecular Dynamics
Demonstration: 1D-Harmonic Oscillator
U(x) =1
2kx2 ) md
2x
dt2= kx
F = dUdx
= kx Analytical solution obtainableReq. two initial conditions:
x(0) x(0)&
Analytical integration of differential equation:
! =
rk
m
x(t) = A cos(!t)
v(t) = Aw sin(!t)
x(0) = A v(0) = 0For the initial conditions:
time(t)
x(t)
; v(t
)
2D-Harmonic OscillatorU(x, y) =
1
2kx2 +
1
2ky2
U(x,y)
x y
U(x,y)
xy
forces direct the system to the minimum PE
Fx = @U@x
= kx Fy = @U@y
= ky
F = Fxi+ Fyj
PE
KE
How about atoms? Classical mechanics? or quantum mechanics? It is a good assumption to treat the atomic
motions classically
atom 2
atom 3
atom 1v1
v2
v3
each degree of freedom of an atom will be having unique
positions, velocities, and
forces
F = md2x
dt2i ii i=1,3N
N=number of particles
Each degree of freedom has an equation of motion (classical) as you have seen for a 2D-harmonic oscillator (before)
For e.g. consider two atoms: 1 2
x1
y1
z1
x2
y2
z2
Fx1 =M1d2x1dt2
Fy1 =M1d2y1dt2
Fz1 =M1d2z1dt2
For atom 1 For atom 2
Fx2 =M2d2x2dt2
Fy2 =M2d2y2dt2
Fz2 =M2d2z2dt2
6 Eq. of motion to
solve independently
The force acting on each degree of freedom of every atom has the information about the inter-atomic interactions
Interatomic interactions are governed by the potential energy surface (as negative of the gradient of the potential energy is the force)
Lennard Jones Potential
ULJ(R1, ,RN ) =XI
XJ>I
4
"
RIJ
12
RIJ
6#
FI = @ULJ(R1, ,RN )@RII = 1, , N
RIJ = |RI RJ |
LJ potential is an example of interatomic interaction
Note: it is a 3D-vector(x,y,z components)
Numerical Integration
VI(t) = VI(0) +
Z t0dFI()
MI
RI(t) = RI(0) +
Z t0
dVI() atom 2 atom 3
atom 1
many body interactions within the
force: numerical integration is required
R
tt
Velocity Verlet Algorithm
RI(t +t) = RI(t) + RI(t)t +1
2RI(t)t
2 +O(t3)
ULJ (R1(t), ,RN (t))
RI(t +t) = RI(t) +t
2
hRI(t) + RI(t +t)
i+O(t2)
ULJ (R1(t+t), ,RN (t+t))
derivat
ive
derivative
deri
vativ
e
Originally by Carl Stoermer (in 1907, particles in electric field)
tt
tt
RI RI
velocity VerletnRI(0), RI(0)
o nRI(t), RI(t)
o nRI(2t), RI(2t)
o
Molecular Dynamics (MD)
trajectory